Trigonometry & Inverse Trigonometry
Tangent Relations
Grade 11
Question:
<p><strong>Statement I:</strong> \(\tan 50° - \tan 30° - \tan 20° = \tan 50° \tan 30° \tan 20°\)</p><p><strong>Statement II:</strong> If \(x = y + z\), then \(\tan x - \tan y - \tan z = \tan x \tan y \tan z\)</p>
<p>(a) A</p>
<p>(b) B</p>
<p>(c) C</p>
<p>(d) D</p>
Step-by-Step Solution
Key Concept: When three angles sum to a specific value, use the tangent addition formula to derive a relationship between the tangents of those angles.
<p><strong>Solution:</strong> Since \(50° = 30° + 20°\), we have:</p><p>\(\tan 50° = \tan(30° + 20°) = \frac{\tan 30° + \tan 20°}{1 - \tan 30° \tan 20°}\)</p><p>This gives us:</p><p>\(\tan 50°(1 - \tan 30° \tan 20°) = \tan 30° + \tan 20°\)</p><p>\(\tan 50° - \tan 50° \tan 30° \tan 20° = \tan 30° + \tan 20°\)</p><p>\(\tan 50° - \tan 30° - \tan 20° = \tan 50° \tan 30° \tan 20°\)</p><p>Statement I is true. Statement II is the general principle on which Statement I is based, so Statement II is also true. The answer is A.</p>
Correct Answer: A